Decomposition Of Complete Lie Color Algebra And Uniqueness

In this thesis,we will study a special king of Lie colour algebra called complete whose centers are zero and all derivations are inner.In section 1 ,some basic concepts of Lie colour algebra will be recalled .The derivation algebra of Lie colour algebra can be Lie colour algebra .Then the solvable Lie algebra and nilpotent Lie algebra will be introduced .At last we give the definition of homomorphism and isomorph between Lie colour algebra .In section 2 ,if one Lie colour algebra can be decomposed to direct sum of two ideas ,the same as the center.And when the center is zero , the derivation and the inner derivation can be also decomposed .Lie colour algebra L is complete if and only if its two ideas are both complete . And then the definition of simply complete Lie colour algebra and L self-homomorphism are given .A Lie colour algebra is simply complete if and only if it can not be decomposed . At last we introduce the most important theory in this thesis which is the uniqueness of decomposition . Complete Lie colour algebra of any finite dimension can be decomposed to direct sum of simply complete ideas .And the decomposition is unique except the order of the ideas .In section 3 ,we will introduce bilinear form B on Lie colour algebra L. (L.B) is called quadratic if B is color symmetric .non-degenerate and invariant .In this case ,B is called an invariant scalar product on L .We also give the definition of the non-degenerate idea . If I is a non-degenerate idea of L ,then is I1- .Then we study that quadratic Lie colour algebra also can be decomposed to direct sum of ideas who contains no nontrivial non-degenerate idea of L . And the decomposition is unique except the order of the ideas .